The researcher randomly selects, without replacement, a subset of items from a
finite population.

Each item in the population can be classified as a success or a failure.

Suppose, for example, that we randomly select 5 cards from an
ordinary deck of playing cards. We might ask: What is the probability of
selecting exactly 3 red cards? In this example, selecting a red card (a heart
or a diamond) would be classified as a success; and selecting a black card (a
club or a spade) would be classified as a failure.

What is a hypergeometric
distribution?

A hypergeometric distribution is a
probability distribution. It refers to the probabilities associated
with the number of successes in a hypergeometric experiment.

For example, suppose we randomly select 5 cards from an ordinary
deck of playing cards. We might ask: What is the probability distribution for
the number of red cards in our selection. In this example, selecting a red card
would be classified as a success. The probabilities associated with each
possible outcome are an example of a hypergeometric distribution, as shown
below.

Outcome

Hypergeometric Probability

Cumulative Probability

0 red cards

0.025

0.025

1 red card

0.150

0.175

2 red cards

0.325

0.500

3 red cards

0.325

0.825

4 red cards

0.150

0.975

5 red cards

0.025

1.00

What is a population size?

In a hypergeometric experiment, a set of items are randomly
selected from a finite population. The total number of items in the population
is the population size.

For example, suppose 5 cards are selected from an ordinary deck
of playing cards. Here, the population size is the total number of cards from
which the selection is made. Since an ordinary deck consists of 52 cards, the
population size would be 52.

What is a sample size?

In a hypergeometric experiment, a set of items are randomly
selected from a finite population. The total number of items selected from the
population is the sample size.

For example, suppose 5 cards are selected from an ordinary deck
of playing cards. Here, the sample size is the total number of cards selected.
Thus, the sample size would be 5.

What is the number of successes?

In a hypergeometric experiment, each element in the population
can be classified as a success or a failure. The number of successes is a count
of the successes in a particular grouping. Thus, the number of successes in the
sample is a count of successes in the sample; and the number of successes in
the population is a count of successes in the population.

What is a hypergeometric probability?

A hypergeometric probability refers to a probability associated
with a hypergeometric experiment. For example, suppose we randomly select 5
cards from an ordinary deck of playing cards. We might ask: What is the
probability of selecting EXACTLY 3 red cards? The probability of getting
EXACTLY 3 red cards would be an example of a hypergeometric probability,
which is indicated by the following notation: P(X = 3).

A cumulative hypergeometric probability refers to a sum of
probabilities associated with a hypergeometric experiment. To compute a
cumulative hypergeometric probability, we may need to add one or more
individual probabilities.

For example, suppose we randomly select 5 cards from an ordinary
deck of playing card. We might ask: What is the probability of selecting AT
MOST 2 red cards? The cumulative probability of getting AT MOST 2 red cards
would be equal to the probability of selecting 0 red cards plus the probability
of selecting 1 red card plus the probability of selecting 2 red cards.
Notationally, this probability would be indicated by P(X < 2).

The table above shows the
cumulative probability for getting at most 2 red cards in a random deal of 5
cards. That probability is 0.500. Thus, P(X < 2) = 0.500.

Hypergeometric Distribution: Sample Problems

Suppose you select randomly select 12 cards without replacement from an
ordinary deck of playing cards. What is the probability that EXACTLY 7 of those
cards will be black (i.e., either a club or spade)?

Solution:

We know the following:

The total population size is 52 (since there are 52 cards in the deck).

The total sample size is 12 (since we are selecting 12 cards).

The number of successes in the population is 26. (Here, we define a success as
choosing a black card, and there are 26 black cards in an ordinary deck of
playing cards.).

The number of successes in the sample is 7 (since there are 7 black cards in
the sample that we select).

Therefore, we plug those numbers into the Hypergeometric
Calculator
and hit the Calculate button. The calculator reports that the hypergeometric
probability is 0.210. That is the probability of getting EXACTLY 7 black cards
in our randomly-selected sample of 12 cards.

The calculator also reports cumulative probabilities.
For example, the probability of getting AT MOST 7 black cards in
our sample is 0.838. That is, P(X < 7) = 0.838.

Suppose we are playing 5-card stud with honest players using a fair deck. What
is the probability that you will be dealt AT MOST 2 aces? (Note: In 5-card
stud, each player is dealt 5 cards.)

Solution:

We know the following:

The total population size is 52 (since there are 52 cards in the full deck).

The total sample size is 5 (since we are dealt 5 cards).

The number of successes in the population is 4 (since there are 4 aces in a
full deck of cards).

The number of successes in the sample is 2 (since we are dealt 2 aces, at
most.).

Therefore, we plug those numbers into the Hypergeometric
Calculator and hit the Calculate button. The calculator reports that
the P(X < 2) is 0.998. That is the probability we
are dealt AT MOST 2 aces. The cumulative probability is the sum of three
probabilities: the probability that we have zero aces, the probability that we
have 1 ace, and the probability that we have 2 aces.

Note that the calculator also displays the hypergeometric probability - the
probability that we have EXACTLY 2 aces. The hypergeometric probability is
0.040.